The Shape of Music

How do harmony and melody combine to make music?

From the seedmagazine.com article:
Roughly 2,500 years ago, Pythagoras observed that objects, such as the anvils he purportedly studied, produced harmonious sounds while vibrating at frequencies in simple whole-number ratios.

More complex ratios gave rise to more dissonant sounds, which indicated that human beings were unconsciously sensitive to mathematical relationships inherent in nature. By showing that the world could be described mathematically, Pythagoras not only provided an important inspiration for physics, but he also discovered a particular affinity between mathematics and music–one that Gottfried Leibniz was to invoke centuries later when he described music as the “unknowing exercise of our mathematical faculties.”

For a thousand years, Western musicians have endeavored to satisfy two fundamental constraints in their compositions. The first is that melodies should, in general, move by short distances. When played on a piano, melodies typically move to nearby keys rather than take large jumps across the keyboard. The second is that music should use chords (collections of simultaneously sounded notes) that are audibly similar. Rather than leap willy-nilly between completely unrelated sonorities, musicians typically restrict themselves to small portions of the musical universe, for instance by using only major and minor chords. While the melodic constraint is nearly universal, the harmonic constraint is more particularly Western: Many non-Western styles either reject chords altogether, using only one note at a time or build entire pieces around a single unchanging harmony.

Together these constraints ensure a two-dimensional coherence in Western music analogous to that of a woven cloth. Music is a collection of simultaneously occurring melodies, parallel horizontal threads that are held together tightly by short-distance motion. But Western music also has a vertical, or harmonic, coherence. If we consider only the notes sounding at any one instant, we find that they form familiar chords related to those that sound at other instants of time. These basic requirements impose nontrivial constraints on composers–not just any sequence of chords we imagine can generate a collection of short-distance melodies. We might therefore ask, how do we combine harmony and melody to make music? In other words, what makes music sound good?

Read the article here.

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